Special Relativity: Real Time Dilation & Length Contraction
Chapter 7 introduced Lorentz and FitzGerald's contraction as a genuine ad hoc fix, invented specifically to rescue the ether. This chapter gives it its real, principled derivation — and closes with three real, independently measured confirmations of the theory that grew from it.
Einstein's Two Postulates (1905)
Einstein's special relativity rests on two deceptively simple starting assumptions:
- The principle of relativity: the laws of physics are identical in every uniformly moving (non-accelerating) reference frame — no experiment performed entirely inside a smoothly moving lab can distinguish it from a stationary one.
- The constancy of the speed of light: light travels at c in a vacuum, regardless of the motion of whatever emitted it or whoever is measuring it.
That second postulate is a direct, real consequence of Chapter 6's own Maxwell's equations, which give a fixed speed for electromagnetic waves with no reference to any observer's own motion at all — and it's precisely this postulate that Chapter 7's own Michelson-Morley null result is genuinely consistent with, even though (as Chapter 7 established) Einstein himself denied that experiment was a significant direct trigger for his own reasoning.
The Lorentz Factor, Properly Derived
From these two postulates alone, a single factor emerges that governs every relativistic effect:
Time Dilation
A moving clock, observed from a stationary frame, runs slow:
Here Δt is the "proper time" — the time interval measured by a clock at rest relative to the event itself — and Δt' is the longer time interval a stationary outside observer measures for that same interval.
Worked Example: Time Dilation at 0.8c
A spacecraft travels at 0.8c. If 10 s pass on the spacecraft's own onboard clock, how much time passes for an observer watching from Earth?
γ = 1/√(1 − 0.64)
γ = 1/√0.36 = 1/0.6 ≈ 1.667
Δt' = 1.667 × 10 ≈ 16.7 s
While 10 s pass aboard the spacecraft, roughly 16.7 s pass for the Earth-based observer — a genuinely real, measurable difference, not merely an illusion of measurement.
Length Contraction
An object moving relative to an observer is measured as shorter, along its own direction of motion, than its "proper length" (the length measured in the object's own rest frame):
Using the same spacecraft (γ ≈ 1.667): a 100 m spacecraft, measured by the Earth-based observer while travelling at 0.8c, would measure only L' = 100/1.667 ≈ 60 m — even though the crew aboard, in their own rest frame, still measures their own ship at the full 100 m.
Real Confirmation 1: Muon Decay
Muons, created when cosmic rays strike the upper atmosphere, decay with a real mean lifetime of just 2.2 microseconds when at rest. Classically, even travelling near light speed, the overwhelming majority should decay before ever reaching the ground — a real 1963 experiment (Frisch and Smith) calculated that only about 27 muons per hour should reach sea level in Cambridge, Massachusetts without time dilation. What they actually measured was roughly 412 muons per hour — a real, striking discrepancy resolved exactly by time dilation, with the muons' own real measured speed (roughly 0.995c) producing a measured dilation factor of 8.8±0.8, closely matching the theoretical prediction.
Real Confirmation 2: The Hafele-Keating Experiment (1971)
Joseph Hafele and Richard Keating flew four real caesium-beam atomic clocks around the world twice aboard commercial airliners — once eastward, once westward — and compared them against a stationary reference clock at the US Naval Observatory upon return.
| Direction | Predicted (ns) | Measured (ns) |
|---|---|---|
| Eastward | −40 ± 23 | −59 ± 10 |
| Westward | +275 ± 21 | +273 ± 7 |
Both real measured results matched the predictions from special and general relativity combined, well within experimental error. The whole experiment, genuinely, cost only about $8,000 — with $7,600 of that spent on the researchers' own plane tickets.
Real Confirmation 3: GPS Satellites
Every GPS satellite, orbiting at roughly 3,874 m/s, experiences real time dilation from its own motion, losing about 7,214 nanoseconds per day relative to a ground clock — but its greater altitude (weaker gravity, a general-relativistic effect this course doesn't derive in depth) causes it to gain roughly 45,850 nanoseconds per day, for a real net gain of about 38.6 microseconds per day.
Three Real Confirmations, Compared
| Confirmation | Real Measured Effect | Scale |
|---|---|---|
| Muon decay | 412 vs. 27 predicted per hour without dilation | Particle physics |
| Hafele-Keating (1971) | Nanosecond-scale clock shifts, matching prediction | Commercial aircraft |
| GPS satellites | 38.6 µs/day net drift, corrected continuously | Everyday technology |
Hands-On Exercises
Quick Reference
- Einstein's two postulates (1905): the relativity principle, and the constancy of the speed of light
- Lorentz factor: γ = 1/√(1 − v²/c²) — the same formula from Chapter 7, now derived rather than assumed
- Time dilation: Δt' = γΔt (moving clocks run slow)
- Length contraction: L' = L/γ (moving objects measure shorter along their motion)
- Real confirmations: muon decay (412 vs. 27/hour), Hafele-Keating (1971, atomic clocks on airliners), and GPS (38.6 µs/day, corrected in every satellite's own clock)